Mix basil, olive oil, garlic, Italian seasoning, and salt in a bowl. Add shrimp and toss to evenly coat. Cover the bowl with plastic wrap and marinate in the refrigerator for 1 hour.
Preheat grill for medium heat and lightly oil the grate. Remove shrimp from the marinade and shake off excess. Discard the remaining marinade.
Cook shrimp on preheated grill until they are bright pink on the outside and the meat is no longer transparent in the center, about 5 minutes.
Bring a large pot of lightly salted water to a boil. Cook angel hair in the boiling water, stirring occasionally, until cooked through but firm to the bite, 4 to 5 minutes. Drain.
Melt butter in a large, nonstick skillet over medium heat; stir in white wine and bring to a boil. Add heavy cream and Parmesan cheese; simmer until reduced, about 10 minutes. Stir in angel hair pasta, tomatoes, and mozzarella cheese. Top with shrimp to serve.
